Wednesday, August 15, 2007
JSH: May have it!
I've been working on the factoring problem with a simple approach I call surrogate factoring where I had a breakthrough yesterday by considering T = p_1*p_2 directly and wondering what would happen if I just picked x, with the governing equations being
x^2 = y^2 mod T
and k = 2x mod T, where T is the target composite to be factored.
I realized that the surrogate I use to factor needed to be roughly equal to T or greater than it, but when you solve everything out as I have I actually put in the freaking thing that x*T is what you need to get near!
What I've usually done is just kind of randomly pick k and worry about the k/T ratio, but you can pick x to force k, with
k = 2x
just using the explicit, and pick a very small x, like x=1, which would then indicate that the surrogate needs to be really close to T, OR if you DO pick k, you need to solve for x, using
x = k*(2)^{-1} mod T
and pick the smallest value to multiply times T to see the minimum size of your surrogate, if I'm not still missing something but that may be the last clue with which 100% factoring may be achieved.!!!
Could it be nearly over? Could I finally have it?
If so, then math people need to get off the namecalling thing and start paying attention to my research!
As I may finally have the factoring problem in hand.
So I guess it's the Hammer.
I've talked a lot about the Hammer over the years and the reference is to Thor's Hammer, as I've always been into mythology and like Norse and Celtic mythology and Thor is a fascinating character.
Note that Thor is HUGE and has red hair and a fiery red beard, not the Goldilocks of modern American comic books.
He was a mythic hero to me as a child. You need heroes in this day and age to keep going, when there seems to be no hope.
That's what heroes are for, to keep going, even when there seems to be no hope.
Thank you Thor and all the gods of the various mythologies…
The Hammer I think is here.
There will be more checking, but I feel greater confidence so I will state more.
I think Mjolnir has arrived…and the earth shakes yet again in recognition.
The gods cheer. Even the giants roar.
And the earth shakes.
[A reply to someone who wrote that James might now try to win the RSA challenge.]
They didn't even drop the money until after I came up with the first part of this approach.
I did that LAST August back in 2006.
Mathematicians had an ENTIRE YEAR to accept that this approach might work and help the world to adjust, but you people seem more intent on running a political campaign against my research.
If this works then I would think that some of you can be charged with treason by your various countries for that and it will not take me demonstrating as if this works then someone will use it.
So yes, the wager now is your life.
Not mine, as I've done everything I can in an impossible situation, but now at this end if you people play the same stupid games as you fight against the progress of the entire human species if it is your life that is the price then you gave it.
YOU gave it.
You cannot make demands of me.
To me you are already traitors—of the human race.
[A reply to someone who, concerning the number 17413590682008709732516359924590332789077936369050, asked James which k would he pick and does it factor.]
That is not a bad question. Answer is k=2. That matches with x = 1 mod T, and should give you the minimum surrogate to factor.
I've been thinking about this idea all day and now realize how crucial it is to know what x is, since the surrogate will roughly be about the size of x*T.
Before I focused on the k/T ratio which gave random behavior and decreasing factoring odds as T increased in size.
Trivial math throughout the proof, but simple as it is, you have to figure it out first to be able to look back and see how simple it is.
Everything is mod T, so you quickly get some serious limits on how hard it can be.
x^2 = y^2 mod T
and k = 2x mod T, where T is the target composite to be factored.
I realized that the surrogate I use to factor needed to be roughly equal to T or greater than it, but when you solve everything out as I have I actually put in the freaking thing that x*T is what you need to get near!
What I've usually done is just kind of randomly pick k and worry about the k/T ratio, but you can pick x to force k, with
k = 2x
just using the explicit, and pick a very small x, like x=1, which would then indicate that the surrogate needs to be really close to T, OR if you DO pick k, you need to solve for x, using
x = k*(2)^{-1} mod T
and pick the smallest value to multiply times T to see the minimum size of your surrogate, if I'm not still missing something but that may be the last clue with which 100% factoring may be achieved.!!!
Could it be nearly over? Could I finally have it?
If so, then math people need to get off the namecalling thing and start paying attention to my research!
As I may finally have the factoring problem in hand.
So I guess it's the Hammer.
I've talked a lot about the Hammer over the years and the reference is to Thor's Hammer, as I've always been into mythology and like Norse and Celtic mythology and Thor is a fascinating character.
Note that Thor is HUGE and has red hair and a fiery red beard, not the Goldilocks of modern American comic books.
He was a mythic hero to me as a child. You need heroes in this day and age to keep going, when there seems to be no hope.
That's what heroes are for, to keep going, even when there seems to be no hope.
Thank you Thor and all the gods of the various mythologies…
The Hammer I think is here.
There will be more checking, but I feel greater confidence so I will state more.
I think Mjolnir has arrived…and the earth shakes yet again in recognition.
The gods cheer. Even the giants roar.
And the earth shakes.
[A reply to someone who wrote that James might now try to win the RSA challenge.]
They didn't even drop the money until after I came up with the first part of this approach.
I did that LAST August back in 2006.
Mathematicians had an ENTIRE YEAR to accept that this approach might work and help the world to adjust, but you people seem more intent on running a political campaign against my research.
If this works then I would think that some of you can be charged with treason by your various countries for that and it will not take me demonstrating as if this works then someone will use it.
So yes, the wager now is your life.
Not mine, as I've done everything I can in an impossible situation, but now at this end if you people play the same stupid games as you fight against the progress of the entire human species if it is your life that is the price then you gave it.
YOU gave it.
You cannot make demands of me.
To me you are already traitors—of the human race.
[A reply to someone who, concerning the number 17413590682008709732516359924590332789077936369050, asked James which k would he pick and does it factor.]
That is not a bad question. Answer is k=2. That matches with x = 1 mod T, and should give you the minimum surrogate to factor.
I've been thinking about this idea all day and now realize how crucial it is to know what x is, since the surrogate will roughly be about the size of x*T.
Before I focused on the k/T ratio which gave random behavior and decreasing factoring odds as T increased in size.
Trivial math throughout the proof, but simple as it is, you have to figure it out first to be able to look back and see how simple it is.
Everything is mod T, so you quickly get some serious limits on how hard it can be.
# posted by James Harris @ 22:34 
